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<a name="Rational-Approximations-1"></a>
<h3 class="section">17.7 Rational Approximations</h3>

<a name="XREFrat"></a><dl>
<dt><a name="index-rat"></a><em><var>s</var> =</em> <strong>rat</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-rat-1"></a><em><var>s</var> =</em> <strong>rat</strong> <em>(<var>x</var>, <var>tol</var>)</em></dt>
<dt><a name="index-rat-2"></a><em>[<var>n</var>, <var>d</var>] =</em> <strong>rat</strong> <em>(&hellip;)</em></dt>
<dd>
<p>Find a rational approximation of <var>x</var> to within the tolerance defined by
<var>tol</var>.
</p>
<p>If unspecified, the default tolerance is <code>1e-6 * norm (<var>x</var>(:), 1)</code>.
</p>
<p>When called with one output argument, return a string containing a
continued fraction expansion (multiple terms).
</p>
<p>When called with two output arguments, return numeric matrices for the
numerator and denominator of a fractional representation of <var>x</var> such
that <code><var>x</var> = <var>n</var> ./ <var>d</var></code>.
</p>
<p>For example:
</p>
<div class="example">
<pre class="example">s = rat (pi)
&rArr; s = 3 + 1/(7 + 1/16)

[n, d] = rat (pi)
&rArr; n =  355
&rArr; d =  113

n / d - pi
&rArr; 0.00000026676
</pre></div>

<p>Programming Note: With one output <code>rat</code> produces a string which is a
continued fraction expansion.  To produce a string which is a simple
fraction (one numerator, one denominator) use <code>rats</code>.
</p>

<p><strong>See also:</strong> <a href="#XREFrats">rats</a>, <a href="Terminal-Output.html#XREFformat">format</a>.
</p></dd></dl>


<a name="XREFrats"></a><dl>
<dt><a name="index-rats"></a><em></em> <strong>rats</strong> <em>(<var>x</var>, <var>len</var>)</em></dt>
<dd><p>Convert <var>x</var> into a rational approximation represented as a string.
</p>
<p>The string can be converted back into a matrix as follows:
</p>
<div class="example">
<pre class="example">r = rats (hilb (4));
x = str2num (r)
</pre></div>

<p>The optional second argument defines the maximum length of the string
representing the elements of <var>x</var>.  By default <var>len</var> is 9.
</p>
<p>If the length of the smallest possible rational approximation exceeds
<var>len</var>, an asterisk (*) padded with spaces will be returned instead.
</p>
<p><strong>See also:</strong> <a href="Terminal-Output.html#XREFformat">format</a>, <a href="#XREFrat">rat</a>.
</p></dd></dl>





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